MATEMATIQKI VESNIK
originalni nauqni rad
research paper
66, 2 (2014), 178–189
June 2014
SET-VALUED PREŠIĆ-ĆIRIĆ TYPE CONTRACTION
IN 0-COMPLETE PARTIAL METRIC SPACES
Satish Shukla
Abstract. The purpose of this paper is to introduce the set-valued Prešić-Ćirić type contraction in 0-complete partial metric spaces and to prove some coincidence and common fixed
point theorems for such mappings in product spaces, in partial metric case. Results of this paper extend, generalize and unify several known results in metric and partial metric spaces. An
example shows how the results of this paper can be used while the existing one cannot.
1. Introduction and preliminaries
There are a number of generalizations of Banach contraction principle. One
such generalization is given by S.B. Prešić [28,29] in 1965. Prešić proved following
theorem.
Theorem 1. Let (X, d) be a complete metric space, k a positive integer and
T : X k → X a mapping satisfying the following contractive type condition:
d(T (x1 , x2 , . . . , xk ), T (x2 , x3 , . . . , xk+1 )) ≤
k
P
i=1
qi d(xi , xi+1 )
(1)
for every x1 , x2 , . . . , xk+1 ∈ X, where q1 , q2 , . . . , qk are nonnegative constants such
that q1 + q2 + · · · + qk < 1. Then there exists a unique point x ∈ X such that
T (x, x, . . . , x) = x. Moreover, if x1 , x2 , . . . , xk are arbitrary points in X and for
n ∈ N, xn+k = T (xn , xn+1 , . . . , xn+k−1 ), then the sequence {xn } is convergent and
lim xn = T (lim xn , lim xn , . . . , lim xn ).
Note that condition (1) in the case k = 1 reduces to the well-known Banach
contraction mapping principle. So, Theorem 1 is a generalization of the Banach
fixed point theorem. Some generalizations and applications of Theorem 1 can be
seen in [11,13,16,19,20,25–27,33,35,36,38–40].
2010 Mathematics Subject Classification: 47H10, 54H25
Keywords and phrases: Set-valued mapping; partial metric space; fixed point; Prešić type
mapping.
178
179
Set-valued Prešić-Ćirić type contraction
Inspired by the results in Theorem 1, Ćirić and Prešić [13] proved following
theorem.
Theorem 2. Let (X, d) be a complete metric space, k a positive integer and
T : X k → X a mapping satisfying the following contractive type condition;
d(T (x1 , x2 , . . . , xk ), T (x2 , x3 , . . . , xk+1 )) ≤ λ max{d(xi , xi+1 ) : 1 ≤ i ≤ k},
where λ ∈ [0, 1) is a constant and x1 , x2 , . . . , xk+1 are arbitrary points in X. Then
there exists a point x in X such that T (x, x, . . . , x) = x. Moreover, if x1 , x2 , . . . , xk
are arbitrary points in X and for n ∈ N, xn+k = T (xn , xn+1 , . . . , xn+k−1 ), then
the sequence {xn } is convergent and lim xn = T (lim xn , lim xn , . . . , lim xn ). If in
addition we suppose that on diagonal 4 ⊂ X k , d(T (u, u, . . . , u), T (v, v, . . . , v)) <
d(u, v) holds for u, v ∈ X, with u 6= v, then x is a unique fixed point satisfying
x = T (x, x, . . . , x).
Nadler [24] generalized the Banach contraction mapping principle to set-valued
functions and proved the following fixed point theorem.
Theorem 3. Let (X, d) be a complete metric space and let T be a mapping
from X into CB(X) (here CB(X) denotes the set of all nonempty closed bounded
subset of X) such that for all x, y ∈ X,
H(T x, T y) ≤ λd(x, y)
where, 0 ≤ λ < 1. Then T has a fixed point.
After the work of Nadler, several authors proved fixed point results for setvalued mappings (see, e.g., [5,6,8,10,12,15,23,39–41]).
Recently, in [39], the author introduced the notion of weak compatibility of
set-valued Prešić type mappings with a single-valued mapping and proved some
coincidence and common fixed point theorems for such mappings in product spaces.
The following theorem was one of the main results of [39].
Theorem 4. Let (X, d) be any complete metric space, k a positive integer.
Let f : X k → CB(X) and g : X → X be two mappings such that g(X) is a closed
subspace of X and f (x1 , x2 , . . . , xk ) ⊂ g(X) for all x1 , x2 , . . . , xk ∈ X. Suppose
that the following condition holds:
H(f (x1 , x2 , . . . , xk ), f (x2 , x3 , . . . , xk+1 )) ≤
k
P
i=1
αi d(gxi , gxi+1 ),
for all x1 , x2 , . . . , xk+1 ∈ X, where αi are nonnegative constants such that
Pk
i=1 αi < 1. Then f and g have a point of coincidence v ∈ X.
The above theorem generalizes the results of Prešić and Nadler in product
spaces in metric case. A generalization of the above theorem can be seen in [40].
On the other hand, Matthews [22] introduced the notion of a partial metric
space as a part of the study of denotational semantics of dataflow networks, with
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S. Shukla
the interesting property of “non-zero self distance” in the space. He showed that
the Banach contraction mapping theorem can be generalized to the partial metric
context for applications in program verification. Subsequently, several authors (see,
e.g., [1–4,6,7,9,14,17,18,30–32,34,37]) derived fixed point theorems in partial metric
spaces. Romaguera [30] introduced the notion of 0-Cauchy sequence, 0-complete
partial metric spaces and proved some characterizations of partial metric spaces in
terms of completeness and 0-completeness.
Recently, Aydi et al. [6] introduced the notion of partial Hausdorff metric and
extended the Nadler’s theorem to partial metric spaces.
In the present paper, we prove some coincidence and common fixed point
theorems for the mappings satisfying Prešić-Ćirić type contractive conditions (see
[13]) in 0-complete partial metric spaces. Our results extend, generalize and unify
the results of Matthews [22], Prešić [28], Ćirić and Prešić [13], Nadler [24] and
recent results of Shukla et al. [39] and Aydi et al. [6] to 0-complete partial metric
spaces.
Consistent with [4,6,18,22,30,32], the following definitions and results will be
needed in the sequel.
Definition 1. A partial metric on a nonempty set X is a function p : X ×X →
R+ (R+ stands for nonnegative reals) such that for all x, y, z ∈ X:
(P1) x = y ⇔ p(x, x) = p(x, y) = p(y, y),
(P2) p(x, x) ≤ p(x, y),
(P3) p(x, y) = p(y, x),
(P4) p(x, y) ≤ p(x, z) + p(z, y) − p(z, z).
A partial metric space is a pair (X, p) such that X is a nonempty set and p is
a partial metric on X.
It is clear that, if p(x, y) = 0, then from (P1) and (P2) x = y. But if x = y,
p(x, y) may not be 0. Also every metric space is a partial metric space, with zero
self distance.
Example 1. If p : R+ × R+ → R+ is defined by p(x, y) = max{x, y}, for all
x, y ∈ R+ , then (R+ , p) is a partial metric space.
Some more examples of partial metric space can be seen in [6,18,22].
Each partial metric on X generates a T0 topology τp on X which has a base
the family of open p-balls {Bp (x, ²) : x ∈ X, ² > 0}, where Bp (x, ²) = {y ∈ X :
p(x, y) < p(x, x) + ²} for all x ∈ X and ² > 0.
Theorem 5. [22]For each partial metric p : X × X → R+ the pair (X, d)
where, d(x, y) = 2p(x, y) − p(x, x) − p(y, y) for all x, y ∈ X, is a metric space.
Here (X, d) is called the induced metric space and d is the induced metric.
In further discussion until unless specified (X, d) will represent the induced metric
space.
Set-valued Prešić-Ćirić type contraction
181
Let (X, p) be a partial metric space.
(1) A sequence {xn } in (X, p) converges to a point x ∈ X if and only if p(x, x) =
limn→∞ p(xn , x).
(2) A sequence {xn } in (X, p) is called Cauchy sequence if there exists (and is
finite) limn,m→∞ p(xn , xm ).
(3) (X, p) is said to be complete if every Cauchy sequence {xn } in X converges
with respect to τp to a point x ∈ X such that p(x, x) = limn,m→∞ p(xn , xm ).
(4) A sequence {xn } in (X, p) is called 0-Cauchy sequence if limn,m→∞ p(xn , xm ) =
0. The space (X, p) is said to be 0-complete if every 0-Cauchy sequence in X
converges with respect to τp to a point x ∈ X such that p(x, x) = 0.
Lemma 1. [22,30,32] Let (X, p) be a partial metric space and {xn } be any
sequence in X.
(i) {xn } is a Cauchy sequence in (X, p) if and only if it is a Cauchy sequence in
metric space (X, d).
(ii) (X, p) is complete if and only if the metric space (X, d) is complete. Furthermore, limn→∞ d(xn , x) = 0 if and only if p(x, x) = limn→∞ p(xn , x) =
limn,m→∞ p(xn , xm ).
(iii) Every 0-Cauchy sequence in (X, p) is Cauchy in (X, d).
(iv) If (X, p) is complete then it is 0-complete.
The converse assertions of (iii) and (iv) do not hold. Indeed the partial metric
space (Q ∩ [0, ∞), p), where Q denotes the set of rational numbers and the partial
metric p is given by p(x, y) = max{x, y}, provides an easy example of a 0-complete
partial metric space which is not complete. It is easy to see that every closed subset
of a 0-complete partial metric space is 0-complete.
Let (X, p) be a partial metric space. Let CB p (X) be the family of all nonempty,
closed and bounded subsets of the partial metric space (X, p), induced by the partial
metric p. Note that closedness is taken in the sense of (X, τp ) (τp is the topology
induced by p) and boundedness is given as follows: A is a bounded subset in (X, p)
if there exist x0 ∈ X and M ≥ 0 such that for all a ∈ A, we have a ∈ Bp (x0 , M ),
that is, p(x0 , a) < p(a, a) + M .
For A, B ∈ CB p (X) and x ∈ X, define
p(x, A) = inf{p(x, a) : a ∈ A},
δp (A, B) = sup{p(a, B) : a ∈ A}.
Lemma 2. [4] Let (X, p) be a partial metric space, A ⊂ X. Then a ∈ A if and
only if p(a, A) = p(a, a).
Proposition 1. [6] Let (X, p) be a partial metric space. For any A, B, C ∈
CB p (X), we have the following:
(i) δp (A, A) = sup{p(a, a) : a ∈ A};
(ii) δp (A, A) ≤ δp (A, B);
(iii) δp (A, B) = 0 implies that A ⊆ B;
(iv) δp (A, B) ≤ δp (A, C) + δp (C, B) − inf c∈C p(c, c).
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S. Shukla
Let (X, p) be a partial metric spaces. For A, B ∈ CB p (X), define
Hp (A, B) = max{δp (A, B), δp (B, A)}.
Proposition 2. [6] Let (X, p) be a partial metric space. For A, B, C ∈
CB p (X), we have
(h1) Hp (A, A) ≤ Hp (A, B);
(h2) Hp (A, B) = Hp (B, A);
(h3) Hp (A, B) ≤ Hp (A, C) + Hp (C, B) − inf c∈C p(c, c).
Corollary 1. [6] Let (X, p) be a partial metric space. For A, B ∈ CB p (X)
the following holds
Hp (A, B) = 0 implies that A = B.
In view of Proposition 2 and Corollary 1, we call the mapping Hp : CB p (X) ×
CB (X) → [0, ∞), a partial Hausdorff metric induced by p.
p
Lemma 3. [6] Let (X, p) be a partial metric space and A, B ∈ CB p (X) and
h > 1. For any a ∈ A there exists b = b(a) ∈ B such that p(a, b) ≤ hHp (A, B).
Definition 2. [39] Let X be a nonempty set, k a positive integer, f : X k → 2X
and g : X → X be mappings.
(a) If x ∈ f (x, . . . , x), then x ∈ X is called a fixed point of f .
(b) An element x ∈ X said to be a coincidence point of f and g if gx ∈ f (x, . . . , x).
(c) If w = gx ∈ f (x, . . . , x), then w is called a point of coincidence of f and g.
(d) If x = gx ∈ f (x, . . . , x), then x is called a common fixed point of f and g.
(e) Mappings f and g are said to be commuting if g(f (x, . . . , x)) = f (gx, . . . , gx)
for all x ∈ X.
(f) Mappings f and g are said to be weakly compatible if gx ∈ f (x, . . . , x) implies
g(f (x, . . . , x)) ⊆ f (gx, . . . , gx).
2. Main results
Theorem 6. Let (X, p) be a 0-complete partial metric space, k a positive
integer. Let f : X k → CB p (X) and g : X → X be two mappings such that g(X)
is a closed subspace of X and f (x1 , x2 , . . . , xk ) ⊂ g(X) for all x1 , x2 , . . . , xk ∈ X.
Suppose following condition holds:
Hp (f (x1 , x2 , . . . , xk ), f (x2 , x3 , . . . , xk+1 )) ≤ λ max{p(gxi , gxi+1 ), 1 ≤ i ≤ k} (2)
for all x1 , x2 , . . . , xk+1 ∈ X, where λ ∈ [0, 1). Then f and g have a point of
coincidence v ∈ X.
Proof. We define a sequence {yn } = {gxn } in X as follows: let x1 , x2 , . . . , xk ∈
X be arbitrary and yn = gxn for n = 1, 2, . . . , k. As f (x1 , . . . , xk ) ∈ CB p (X)
and f (x1 , . . . , xk ) ⊂ g(X), we can assume yk+1 = gxk+1 ∈ f (x1 , . . . , xk ), for
Set-valued Prešić-Ćirić type contraction
183
√
some xk+1 ∈ X also, λ < 1 so using Lemma 3 with h = 1/ λ, there exists
yk+2 = gxk+2 ∈ f (x2 , . . . , xk+1 ) such that
p(yk+1 , yk+2 ) = p(gxk+1 , gxk+2 )
1
≤ √ Hp (f (x1 , . . . , xk ), f (x2 , . . . , xk+1 ))
λ
√
≤ λ max{p(gxi , gxi+1 ), 1 ≤ i ≤ k}
√
= λ max{p(yi , yi+1 ), 1 ≤ i ≤ k}.
Similarly, there exists yk+3 = gxk+3 ∈ f (x3 , . . . , xk+2 ) such that
p(yk+2 , yk+3 ) = p(gxk+2 , gxk+3 )
1
≤ √ Hp (f (x2 , . . . , xk+1 ), f (x3 , . . . , xk+2 ))
λ
√
≤ λ max{p(yi , yi+1 ), 2 ≤ i ≤ k + 1}.
Continuing this procedure we obtain a sequence {yn } such that yn = gxn for
n = 1, 2, . . . , k and yn+k = gxn+k ∈ f (xn , . . . , xn+k−1 ) for n = 1, 2, . . . with
√
p(yk+n , yk+n+1 ) ≤ λ max{p(yi , yi+1 ), n ≤ i ≤ n + k − 1}.
(3)
for all n ∈ N.
Set pn = p(gxn , gxn+1 ) = p(yn , yn+1 ) for all n ∈ N and
p(gx1 , gx2 ) p(gx2 , gx3 )
p(gxk , gxk+1 )
µ = max{
,
,... ,
}
2
δ
δ
δk
p1 p2
pk
= max{ , 2 , . . . , k }
δ δ
δ
where δ = λ1/2k . By the method of mathematical induction we shall prove that
pn ≤ µδ n for all n ∈ N.
(4)
By the definition of µ it is clear that (4) is true for n = 1, 2, . . . , k. Let the
k inequalities pn ≤ µδ n , pn+1 ≤ µδ n+1 . . . . , pn+k−1 ≤ µδ n+k−1 be the induction
hypothesis. Using (3) we obtain
pn+k = p(yn+k , yn+k+1 )
√
≤ λ max{p(yi , yi+1 ), n ≤ i ≤ n + k − 1}
√
= λ max{pi , n ≤ i ≤ n + k − 1}
√
= λ max{pn , pn+1 , . . . , pn+k−1 }
√
≤ λ max{µδ n , µδ n+1 , . . . , µδ n+k−1 }
√
= λµδ n
(as δ = λ1/2k < 1)
= µδ n+k .
Thus, inductive proof of (4) is complete. Now we shall show that the sequence
{yn } = {gxn } is a Cauchy sequence in g(X). Let m, n ∈ N with m > n, then using
(4) we obtain
p(yn , ym ) ≤ p(yn , yn+1 ) + p(yn+1 , yn+2 ) + · · · + p(ym−1 , ym )
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S. Shukla
− [p(yn+1 , yn+1 ) + p(yn+2 , yn+2 ) + · · · + p(ym−1 , ym−1 )]
≤ pn + pn+1 + · · · + pm−1
As δ = λ1/2k
that
≤ µδ n + µδ n+1 + · · · + µδ m−1
µδ n
≤ µδ n [1 + δ + δ 2 + · · · ] =
.
1−δ
n
µδ
< 1, therefore 1−δ
→ 0 as n → ∞. So, it follows from above inequality
lim p(yn , ym ) = 0.
n,m→∞
Therefore {yn } = {gxn } is a 0-Cauchy sequence in g(X). As g(X) is closed, there
exists u, v ∈ X such that v = gu and
lim p(yn , v) = lim p(yn , ym ) = p(gu, gu) = p(v, v) = 0.
(5)
n→∞
n,m→∞
We shall show that u is a coincidence point of f and g.
Note that, gxn+k = yn+k ∈ f (xn , xn+1 , . . . , xn+k−1 ) so, for any n ∈ N we have
p(v, f (u, . . . , u)) ≤ p(v, yn+k ) + p(yn+k , f (u, . . . , u))
≤ p(v, yn+k ) + Hp (f (xn , . . . , xn+k−1 ), f (u, . . . , u))
≤ p(v, yn+k ) + Hp (f (xn , . . . , xn+k−1 ), f (xn+1 , . . . , xn+k−1 , u))
+ Hp (f (xn+1 , . . . , xn+k−1 , u), f (xn+2 , . . . , xn+k−1 , u, u))
+ · · · + Hp (f (xn+k−1 , u, . . . , u), f (u, . . . , u)),
and using (2) in the above inequality we obtain
p(v, f (u, . . . , u)) ≤ p(v, yn+k ) + λ max{pn , . . . , pn+k−2 , p(gxn+k−1 , gu)}
+ λ max{pn+1 , . . . , pn+k−2 , p(gxn+k−1 , gu), p(gu, gu)}
+ · · · + λ max{p(gxn+k−1 , gu), p(gu, gu), . . . , p(gu, gu)}
= p(v, yn+k ) + λ max{pn , . . . , pn+k−2 , p(yn+k−1 , v)}
+ λ max{pn+1 , . . . , pn+k−2 , p(yn+k−1 , v), p(v, v)}
+ · · · + λ max{p(yn+k−1 , v), p(v, v), . . . , p(v, v)}.
In view of (5), it follows from the above inequality that p(v, f (u, . . . , u)) = 0 =
p(v, v). As f (u, . . . , u) ∈ CB p (X), by Lemma 2 we have v = gu ∈ f (u, . . . , u) i.e.
u is a coincidence point and v is a point of coincidence of f and g.
Remark 1. If we take p = d, i.e., if we replace partial metric by metric in the
above theorem, we obtain a generalization of the result of [35] in metric spaces.
Taking g = IX in Theorem 6, we obtain the following fixed point result for
set-valued Prešić-Ćirić type contraction.
Corollary 2. Let (X, p) be a 0-complete metric space, k a positive integer.
Let f : X k → CB p (X) be a set-valued Prešić-Ćirić type contraction, i.e., let it
satisfy the following contractive type condition
Hp (f (x1 , x2 , . . . , xk ), f (x2 , x3 , . . . , xk+1 )) ≤ λ max{p(xi , xi+1 ), 1 ≤ i ≤ k} (6)
for all x1 , x2 , . . . , xk+1 ∈ X, where λ ∈ [0, 1). Then f has a fixed point v ∈ X.
Set-valued Prešić-Ćirić type contraction
185
Remark 2. The above corollary is a set-valued version and generalization of
the result of Prešić and Ćirić [13] for set-valued mappings in 0-complete partial
metric spaces. Note that for k = 1 the above corollary reduces to the of result of
Aydi et al. (see Theorem 3.2 of [6]), therefore it is a generalization of the result
of Aydi et al. Also, it generalizes the result of Prešić (Theorem 1) for set-valued
mappings.
The following theorem provides some sufficient conditions for the uniqueness
of point of coincidence of mappings f and g.
Theorem 7. Let (X, p) be a 0-complete partial metric space, k a positive
integer. Let f : X k → CB p (X) and g : X → X be two mappings such that, all the
conditions of Theorem 6 are satisfied and for any coincidence point u of f and g
we have f (u, . . . , u) = {gu}. If
(i) on the diagonal 4 ⊂ X k ,
Hp (f (x, . . . , x), f (y, . . . , y)) < p(gx, gy)
holds for all x, y ∈ X with x 6= y, or
(ii) in condition (2) the constant λ ∈ (0, k1 ).
Then, there exists a unique point of coincidence of f and g. Suppose in addition
that f and g are weakly compatible, then f and g have a unique common fixed point.
Proof. The existence of coincidence point u and point of coincidence v = gu
follows from Theorem 6.
First, suppose that (i) is satisfied. We shall show that the point of coincidence
v is unique. If v 0 is another point of coincidence with coincidence point u0 of f and
g, then f (u0 , . . . , u0 ) = {gu0 } = {v 0 } and we have
p(v, v 0 ) = Hp ({v}, {v 0 })
= Hp (f (u, . . . , u), f (u0 , . . . , u0 ))
< p(gu, gu0 ) = p(v, v 0 ),
a contradiction. So, the point of coincidence of f and g is unique.
Suppose (ii) is satisfied, then using (2) we obtain
p(v, v 0 ) = Hp ({v}, {v 0 })
= Hp (f (u, . . . , u), f (u0 , . . . , u0 ))
≤ Hp (f (u, . . . , u), f (u, . . . , u, u0 )) + Hp (f (u, . . . , u, u0 ), f (u, . . . , u, u0 , u0 ))
+ · · · + Hp (f (u, u0 , . . . , u0 ), f (u0 , . . . , u0 ))
≤ λ max{p(gu, gu), . . . , p(gu, gu), p(gu, gu0 )}
+ λ max{p(gu, gu), . . . , p(gu, gu), p(gu, gu0 ), p(gu0 , gu0 )}
+ · · · + λ max{p(gu, gu0 ), p(gu0 , gu0 ) . . . , p(gu0 , gu0 )}
= kλp(gu, gu0 ) = kλp(v, v 0 ) < p(v, v 0 ),
again a contradiction. So, the point of coincidence of f and g is unique.
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S. Shukla
Suppose that f and g are weakly compatible. Then we have
g(f (u, . . . , u)) ⊆ f (gu, . . . , gu) = f (v, . . . , v) i.e. {gv} ⊆ f (v, . . . , v).
Therefore gv ∈ f (v, . . . , v), which shows that gv is another point of coincidence
of f and g and by uniqueness we have v = gv ∈ f (v, . . . , v). Thus v is a unique
common fixed point of f and g.
The following is a simple example of set-valued Prešić-Ćirić contraction which
illustrate the case when the results of this paper can be used while the existing one
cannot.
Example 2. Let X = Q∩[0, 1] be endowed with the partial metric p : X×X →
R+ defined by
p(x, y) = |x − y| + max{x, y} for all x, y ∈ X.
First, we shall show that the space (X, p) is 0-complete. If {xn } is any 0-Cauchy
sequence in X, then limn,m→∞ p(xn , xm ) = 0, i.e.,
lim [|xn − xm | + max{xn , xm }] = 0.
n,m→∞
(7)
Note that the partial metric space (X, p1 ) is 0-complete, where p1 (x, y) = max{x, y}
for all x, y ∈ X (see [30]). Therefore it follows from (7) that limn→∞ p1 (xn , 0) =
0 = p1 (0, 0) and limn→∞ |xn − 0| = 0. So we have limn→∞ p(xn , 0) = 0 = p(0, 0).
As 0 ∈ X, the space (X, p) is 0-complete.
Note that, the metric d induced by p is given by
d(x, y) = 2|x − y| + 2 max{x, y} − x − y = 3|x − y| for all x, y ∈ X,
and the metric space (X, d) is not complete, and so the partial metric space (X, p)
is not complete. Note that, if x ∈ X then the singleton subset {x} of X is a closed
subset with respect to p. Indeed, for any y ∈ X, we have
y ∈ {x} ⇔ p(y, y) = p(y, {x})
⇔ p(y, y) = p(y, x)
⇔ y = |y − x| + max{y, x}
⇔ y = x.
Thus {x} is closed. Now, for k = 2, define a mapping T : X 2 → X by
½
0,
if x = y = 1;
T (x, y) = x+y
10 , otherwise,
and a mapping f : X 2 → CB p (X) by
f (x, y) = {T (x, y)} ∪ {0} for all x, y ∈ [0, 1].
We shall show that f satisfies condition (6) of Corollary 2 with λ ∈ [ 25 , 1).
If x1 , x2 , x3 ∈ [0, 1) with x1 ≤ x2 ≤ x3 then
x1 + x2
x2 + x3
Hp (f (x1 , x2 ), f (x2 , x3 )) = Hp ({
, 0}, {
, 0})
10
10
Set-valued Prešić-Ćirić type contraction
187
2x3 + x2 − x1 2x1 + 2x2
,
},
10
10
2x3 + x2 − x1 2x2 + 2x3
inf{
,
}}
10
10
1
2x3 + x2 − x1
=
=
(2x3 − x2 + 2x2 − x1 )
10
10
1
≤ max{2x2 − x1 , 2x3 − x2 }
5
1
= max{p(x1 , x2 ), p(x2 , x3 )}.
5
= max{inf{
Therefore (6) is satisfied with λ ∈ [ 52 , 1).
If x1 , x2 , x3 ∈ [0, 1) with x3 ≤ x1 ≤ x2 then
x1 + x2
x2 + x3
, 0}, {
, 0})
10
10
2x1 + x2 − x3 2x1 + 2x2
= max{inf{
,
},
10
10
2x1 + x2 − x3 2x2 + 2x3
inf{
,
}}
10
10
2x1 + x2 − x3
=
10
Hp (f (x1 , x2 ), f (x2 , x3 )) = Hp ({
and max{p(x1 , x2 ), p(x2 , x3 )} = max{2x2 − x1 , 2x2 − x3 } = 2x2 − x3 . Therefore,
(6) is satisfied with λ ∈ [ 25 , 1).
Similarly, if x1 , x2 , x3 ∈ [0, 1) with x2 ≤ x3 ≤ x1 or any one of x1 , x2 , x3 is
equal to 1, then with a similar process one can verify (6).
If any two of x1 , x2 , x3 is equal to 1, e.g., let x1 = x2 = 1 and x3 ∈ [0, 1), then
1 + x3
, 0})
10
2 + 2x3
2 + 2x3
= max{0,
}=
10
10
Hp (f (x1 , x2 ), f (x2 , x3 )) = Hp ({0}, {
and max{p(x1 , x2 ), p(x2 , x3 )} = max{1, 2 − x3 } = 2 − x3 . Therefore, (6) is satisfied
with λ ∈ [ 25 , 1). Similarly, (6) is satisfied in all possible cases with λ ∈ [ 25 , 1). Thus,
all the conditions of Corollary 2 are satisfied and f has a fixed point 0 ∈ X.
On the other hand, as (X, du ) and (X, d) (where du is usual and d is the
induced metric on X) are not complete spaces, we cannot conclude the existence of
fixed point of f with the metric version of Corollary 2. Also, it is easy to see that
f is not a Prešić-Ćirić contraction in both the spaces (X, du ) and (X, d). Indeed,
9
at x1 = x2 = 1, x3 = 10
the mapping f fails to be a Prešić-Ćirić contraction in
these metric spaces. Thus we can say that the class of Prešić-Ćirić contractions in
partial metric spaces is wider than that in metric spaces.
Acknowledgement. The author is grateful to Professor Stojan Radenović
and the referees for their valuable comments and suggestions on this paper.
188
S. Shukla
REFERENCES
[1] T. Abdeljawad, Fixed points of generalized weakly contractive mappings in partial metric
spaces, Math. Comput. Model. 54 (2011), 2923–2927.
[2] T. Abdeljawad, E. Karapinar, K. Taş, Existence and uniqueness of a common fixed point on
partial metric spaces, Appl. Math. Lett. 24 (2011), 1900–1904.
[3] A.G.B. Ahmad, Z.M. Fadail, V.Ć. Rajić, S. Radenović, Nonlinear contractions in 0-complete
partial metric spaces, Abstract Appl. Anal. 2012, Article ID 451239, 13 p., (2012).
[4] I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology
Appl. 157 (2010) 2778–2785.
[5] H. Aydi, M. Abbas, M. Postolache, Coupled coincidence points for hybrid pair of mappings
via mixed monotone property, J. Adv. Math. Stud. 5 (2012), 118–126.
[6] H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler’s fixed point theorem on
partial metric spaces, Topology Appl. 159 (2012), 3234–3242.
[7] C. Di Bari, Z. Kadelburg, H.K. Nashine, S. Radenović, Common fixed points of g-quasicontractions and related mappings in 0-complete partial metric spaces, Fixed Point Theory
Appl. 2012:113 (2012).
[8] M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math.
Anal. Appl. 326 (2007), 772–782.
[9] M. Bukatin, R. Kopperman, S. Matthews, H. Pajoohesh, Partial metric spaces, Am. Math.
Mon. 116 (2009), 708–718.
[10] P. Chaipunya, C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for multivalued mappings in modular metric spaces, Abstr. Appl. Anal. 2012, Article ID 503504,
14 p. (2012).
[11] Y.Z. Chen, A Prešić type contractive condition and its applications, Nonlinear Anal. 71
(2009), 2012–2017.
[12] S.H. Cho, J.S. Bae, Fixed point theorems for multi-valued maps in cone metric spaces, Fixed
Point Theory Appl. 2011:87 (2011).
[13] Lj.B. Ćirić, S.B. Prešić, On Prešić type generalisation of Banach contraction principle, Acta.
Math. Univ. Com. 76 (2007), 143–147.
[14] Lj.B. Ćirić, B. Samet, H. Aydi, C. Vetro, Common fixed points of generalized contractions
on partial metric spaces and an application, Appl. Math. Comput. 218 (2011), 2398–2406.
[15] G.M. Eshaghi, H. Baghani, H. Khodaei, M. Ramezani, A generalization of Nadler’s fixed
point theorem, J. Nonlinear Sci. Appl. 3 (2010), 148–151.
[16] R. George, K.P. Reshma, R. Rajagopalan, A generalized fixed point theorem of Prešić type
in cone metric spaces and application to Markov process, Fixed Point Theory Appl. 2011:85
(2011).
[17] D. Ilić, V. Pavlović, V. Rakočević, Extensions of Zamfirescu theorem to partial metric spaces,
Math. Comput. Model. 55 (2012), 801–809.
[18] Z. Kadelburg, H.K. Nashine, S. Radenović, Fixed point results under various contractive
conditions in partial metric spaces, Rev. Real Acad. Cienc. Exac., Fis. Nat., Ser. A, Mat.
107 (2013), 241–256.
[19] M.S. Khan, M. Berzig, B. Samet, Some convergence results for iterative sequences of Prešić
type and applications, Adv. Difference Equ. 2012:38 (2012).
[20] S.K. Malhotra, S. Shukla, R. Sen, A generalization of Banach contraction principle in ordered
cone metric spaces, J. Adv. Math. Stud. 5 (2012), 59–67.
[21] S.K. Malhotra, S. Shukla, R. Sen, Some coincidence and common fixed point theorems for
Prešić-Reich type mappings in cone metric spaces, Rend. Sem. Mat. Univ. Pol. Torino, 70
(2013) (to appear).
[22] S.G. Matthews, Partial metric topology, In: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci., vol. 728, (1994) pp. 183–197.
[23] N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete
metric spaces, J. Math. Anal. Appl. 141 (1989), 177–188.
Set-valued Prešić-Ćirić type contraction
189
[24] S.B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488.
[25] M. Pǎcurar, A multi-step iterative method for approximating common fixed points of PrešićRus type operators on metric spaces, Studia Univ. “Babeş-Bolyai”, Mathematica, 15 (2010).
[26] M. Pǎcurar, Approximating common fixed points of Prešić-Kannan type operators by a multistep iterative method, An. Şt. Univ. Ovidius Constanţa 17 (2009), 153–168.
[27] M. Pǎcurar, Common fixed points for almost Prešić type operators, Carpathian J. Math. 28
(2012), 117–126.
[28] S.B. Prešić, Sur la convergence des suites, Compt. Rendus Acad. Paris 260 (1965), 3828–
3830.
[29] S.B. Prešić, Sur une classe d’inequations aux differences finite et sur la convergence de
certaines suites, Publ. Inst. Math. Belgrade, 5(19) (1965), 75–78.
[30] S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed
Point Theory Appl. 2010, Article ID 493298, 6 p. (2010).
[31] S. Romaguera, Fixed point theorems for generalized contractions on partial metric spaces,
Topology Appl. 159 (2012), 194–199.
[32] S. Romaguera, Matkowski’s type theorems for generalized contractions on (ordered) partial
metric spaces, Appl. Gen. Topology 12 (2011), 213–220.
[33] S. Shukla, Prešić type results in 2-Banach spaces, Afrika Mat. (2013) DOI 10.1007/s13370013-0174-2.
[34] S. Shukla, I. Altun, R. Sen, Fixed point theorems and asymptotically regular mappings in
partial metric spaces, ISRN Computat. Math. 2013, Article ID 602579, 6 p. (2013).
[35] S. Shukla, B. Fisher, A generalization of Prešić type mappings in metric-like spaces, J.
Operators 2013, Article ID 368501, 5 p. (2013).
[36] S. Shukla, S. Radenović, A generalization of Prešić type mappings in 0-complete ordered
partial metric spaces, Chinese J. Math. 2013, Article ID 859531, 8 p. (2013).
[37] S. Shukla, S. Radenović, Some common fixed point theorems for F-contraction type mappings
in 0-complete partial metric spaces, J. Mathematics 2013, Article ID 878730, 7 p. (2013).
[38] S. Shukla, S. Radenović, S. Pantelić, Some fixed point theorems for Prešić-Hardy-Rogers type
contractions in metric spaces, J. Mathematics 2013, Article ID 295093, 8 p. (2013).
[39] S. Shukla, R. Sen, S. Radenović, Set-valued Prešić type contraction in metric spaces, An.
Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) (2013) (to appear).
[40] S. Shukla, R. Sen, Set-valued Prešić-Reich type mappings in metric spaces, Rev. Real Acad.
Cienc. Exac., Fis. Nat., Ser. A, Mat. DOI 10.1007/s13398-012-0114-2.
[41] D. Wardowski, On set-valued contractions of Nadler type in cone metric spaces, Appl. Math.
Lett. 24 (2011), 275–278.
(received 19.09.2012; in revised form 27.05.2013; available online 15.10.2013)
Department of Applied Mathematics, Shri Vaishnav Institute of Technology & Science, Gram
Baroli Sanwer Road, Indore (M.P.) 453331, India
E-mail: satishmathematics@yahoo.co.in